A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition

Abstract

Abstract A new constructive method for the finite-difference solution of the Laplace equation with the integral boundary condition is proposed and justified. In this method, the approximate solution of the given problem is defined as a sequence of 9-point solutions of the local Dirichlet problems. It is proved that when the exact solution u(x,y) $u(x,y)$ belongs to the Hölder calsses C4,λ $C^{4,\lambda }$, 0<λ<1 $0<\lambda <1$, on the closed solution domain, the uniform estimate of the error of the approximate solution is of order O(h4) $O(h^{4})$, where h is the mesh step. Numerical experiments are given to support analysis made.

Keywords

• Finite difference method
• Nonlocal integral boundary condition
• Laplace’s equation
• Uniform error